answer：Okay, so I have this function f(x) = |3x - 2| + |x - 2|, and I need to solve two parts. First, solve the inequality f(x) ≤ 8, and second, find the range of m such that f(x) is always greater than or equal to (m² - m + 2)|x| for any x. Hmm, let me start with part I.For part I, solving f(x) ≤ 8. Since f(x) is the sum of two absolute value functions, I think I need to consider different cases based on the critical points where the expressions inside the absolute values change their signs. The expressions inside the absolute values are 3x - 2 and x - 2. So, 3x - 2 = 0 when x = 2/3, and x - 2 = 0 when x = 2. So, these are the critical points. Therefore, I can break down the real line into three intervals: x < 2/3, 2/3 ≤ x < 2, and x ≥ 2.Let me handle each interval one by one.1. When x < 2/3: In this interval, 3x - 2 is negative because x < 2/3, so |3x - 2| = -(3x - 2) = 2 - 3x. Similarly, x - 2 is negative because x < 2, so |x - 2| = -(x - 2) = 2 - x. Therefore, f(x) = (2 - 3x) + (2 - x) = 4 - 4x. So, the inequality becomes 4 - 4x ≤ 8. Let me solve this: 4 - 4x ≤ 8 Subtract 4 from both sides: -4x ≤ 4 Divide both sides by -4 (remember to reverse the inequality sign): x ≥ -1. But in this interval, x < 2/3, so combining these, we get -1 ≤ x < 2/3.2. When 2/3 ≤ x < 2: Here, 3x - 2 is non-negative because x ≥ 2/3, so |3x - 2| = 3x - 2. However, x - 2 is still negative because x < 2, so |x - 2| = 2 - x. Therefore, f(x) = (3x - 2) + (2 - x) = 2x. So, the inequality becomes 2x ≤ 8. Solving this: x ≤ 4. But in this interval, x is between 2/3 and 2, so combining these, we get 2/3 ≤ x < 2.3. When x ≥ 2: Both 3x - 2 and x - 2 are non-negative because x ≥ 2, so |3x - 2| = 3x - 2 and |x - 2| = x - 2. Therefore, f(x) = (3x - 2) + (x - 2) = 4x - 4. The inequality becomes 4x - 4 ≤ 8. Solving this: 4x - 4 ≤ 8 Add 4 to both sides: 4x ≤ 12 Divide by 4: x ≤ 3. But in this interval, x ≥ 2, so combining these, we get 2 ≤ x ≤ 3.Now, combining all three intervals, the solution set is from -1 to 3. So, the solution is x ∈ [-1, 3].Okay, that was part I. Now, moving on to part II.Part II says that for any x, f(x) ≥ (m² - m + 2)|x| always holds. We need to find the range of real number m.So, f(x) = |3x - 2| + |x - 2|, and we need this to be greater than or equal to (m² - m + 2)|x| for all x.First, let me note that m² - m + 2 is a quadratic in m. Let me see if it's always positive or not. The discriminant is (-1)² - 4*1*2 = 1 - 8 = -7, which is negative, so m² - m + 2 is always positive for all real m. That's good because we don't have to worry about the coefficient being zero or negative.So, the inequality is f(x) ≥ k|x|, where k = m² - m + 2, and k is positive.We need this to hold for all x. So, for all x, |3x - 2| + |x - 2| ≥ k|x|.I think to find the maximum possible k such that this inequality holds for all x, we can consider the minimum value of f(x)/|x| over all x ≠ 0, and set k to be less than or equal to that minimum.So, let me define g(x) = f(x)/|x| = (|3x - 2| + |x - 2|)/|x|.We need to find the minimum value of g(x) for x ≠ 0, because k must be less than or equal to this minimum to satisfy the inequality for all x.So, let's compute g(x):g(x) = (|3x - 2| + |x - 2|)/|x|Let me consider different cases for x.First, x > 0 and x ≠ 0.Case 1: x > 0.Subcases based on the critical points 2/3 and 2.Subcase 1a: 0 < x < 2/3.In this interval, 3x - 2 is negative, so |3x - 2| = 2 - 3x.x - 2 is negative, so |x - 2| = 2 - x.So, g(x) = (2 - 3x + 2 - x)/x = (4 - 4x)/x = 4/x - 4.We can analyze this function. As x approaches 0 from the right, 4/x approaches infinity, so g(x) approaches infinity. As x approaches 2/3 from the left, g(x) approaches 4/(2/3) - 4 = 6 - 4 = 2.So, in this subcase, g(x) ranges from 2 to infinity.Subcase 1b: 2/3 ≤ x < 2.Here, 3x - 2 is non-negative, so |3x - 2| = 3x - 2.x - 2 is negative, so |x - 2| = 2 - x.Thus, g(x) = (3x - 2 + 2 - x)/x = (2x)/x = 2.So, in this subcase, g(x) is constant 2.Subcase 1c: x ≥ 2.Here, both 3x - 2 and x - 2 are non-negative.So, |3x - 2| = 3x - 2, |x - 2| = x - 2.Thus, g(x) = (3x - 2 + x - 2)/x = (4x - 4)/x = 4 - 4/x.As x approaches 2 from the right, g(x) approaches 4 - 4/2 = 4 - 2 = 2.As x approaches infinity, g(x) approaches 4 - 0 = 4.So, in this subcase, g(x) ranges from 2 to 4.Case 2: x < 0.Here, |x| = -x, so let's compute g(x):g(x) = (|3x - 2| + |x - 2|)/(-x).Let me analyze the expressions inside the absolute values.3x - 2: Since x < 0, 3x is negative, so 3x - 2 is negative. Therefore, |3x - 2| = -(3x - 2) = -3x + 2.x - 2: x is negative, so x - 2 is negative. Therefore, |x - 2| = -(x - 2) = -x + 2.Thus, g(x) = (-3x + 2 - x + 2)/(-x) = (-4x + 4)/(-x) = (4x - 4)/x = 4 - 4/x.Wait, let me double-check that:Wait, (-3x + 2 - x + 2) = (-4x + 4). Then, divided by (-x):(-4x + 4)/(-x) = (4x - 4)/x = 4 - 4/x.Yes, that's correct.So, g(x) = 4 - 4/x.But x is negative here, so let's see:As x approaches 0 from the left, -4/x approaches positive infinity, so 4 - 4/x approaches negative infinity. Wait, that can't be right because g(x) is f(x)/|x|, which is always positive.Wait, maybe I made a mistake in the calculation.Wait, let me recast it:g(x) = (|3x - 2| + |x - 2|)/|x|.For x < 0:|3x - 2| = |3x - 2| = since x < 0, 3x is negative, so 3x - 2 is negative, so |3x - 2| = -(3x - 2) = -3x + 2.Similarly, |x - 2| = |x - 2| = since x < 0, x - 2 is negative, so |x - 2| = -(x - 2) = -x + 2.Therefore, f(x) = (-3x + 2) + (-x + 2) = -4x + 4.So, g(x) = (-4x + 4)/|x|.But x is negative, so |x| = -x.Thus, g(x) = (-4x + 4)/(-x) = (4x - 4)/(-x) = (-4x + 4)/(-x) = (4 - 4x)/(-x) = -4 + 4/x.Wait, that seems different. Let me compute step by step:(-4x + 4)/(-x) = (-4x)/(-x) + 4/(-x) = 4 - 4/x.Yes, that's correct.So, g(x) = 4 - 4/x.But x is negative, so 4/x is negative, so 4 - 4/x is 4 plus a positive number, so it's greater than 4.As x approaches 0 from the left, 4/x approaches negative infinity, so -4/x approaches positive infinity, so g(x) approaches 4 + infinity = infinity.As x approaches negative infinity, 4/x approaches 0, so g(x) approaches 4 - 0 = 4.So, in this case, g(x) is greater than 4 for x < 0.Therefore, combining all cases:For x > 0:- When 0 < x < 2/3: g(x) ranges from 2 to infinity.- When 2/3 ≤ x < 2: g(x) is 2.- When x ≥ 2: g(x) ranges from 2 to 4.For x < 0: g(x) is greater than 4.So, the minimum value of g(x) occurs at x = 2/3 and x = 2, where g(x) = 2. For x > 2/3, g(x) is at least 2, and for x < 0, g(x) is greater than 4.Therefore, the minimum value of g(x) over all x ≠ 0 is 2.Thus, to satisfy f(x) ≥ k|x| for all x, we need k ≤ 2.But k = m² - m + 2, so:m² - m + 2 ≤ 2Simplify:m² - m ≤ 0Factor:m(m - 1) ≤ 0So, this inequality holds when m is between 0 and 1, inclusive.Because the product of m and (m - 1) is less than or equal to zero when one is non-positive and the other is non-negative. So, m ∈ [0, 1].Therefore, the range of m is 0 ≤ m ≤ 1.Let me just verify this.If m = 0: m² - m + 2 = 0 - 0 + 2 = 2.If m = 1: m² - m + 2 = 1 - 1 + 2 = 2.For m between 0 and 1, m² - m + 2 is less than or equal to 2.Yes, that makes sense.So, the range of m is [0, 1].**Final Answer**(I) The solution set is boxed{[-1, 3]}.(II) The range of ( m ) is boxed{[0, 1]}.

answer：Alright, so I have this inequality to solve: (frac{x^2 + 2x^3 - 3x^4}{x + 2x^2 - 3x^3} ge -1). Hmm, okay. Let me try to figure this out step by step.First, I notice that both the numerator and the denominator are polynomials. Maybe I can factor them to simplify the expression. Let me start with the numerator: (x^2 + 2x^3 - 3x^4). I can factor out an (x^2) from all the terms, which gives me (x^2(1 + 2x - 3x^2)). Now, looking at the denominator: (x + 2x^2 - 3x^3). I can factor out an (x) from each term, resulting in (x(1 + 2x - 3x^2)). So, now the expression becomes:[frac{x^2(1 + 2x - 3x^2)}{x(1 + 2x - 3x^2)}]I see that both the numerator and the denominator have a common factor of (1 + 2x - 3x^2). I can cancel this out, but I have to remember that (1 + 2x - 3x^2) cannot be zero because that would make the denominator zero, which is undefined. So, I need to find the values of (x) where (1 + 2x - 3x^2 = 0).Let me solve that quadratic equation: (1 + 2x - 3x^2 = 0). To make it easier, I'll rewrite it as (-3x^2 + 2x + 1 = 0). Multiplying both sides by (-1) gives (3x^2 - 2x - 1 = 0). Using the quadratic formula, (x = frac{2 pm sqrt{(-2)^2 - 4(3)(-1)}}{2(3)} = frac{2 pm sqrt{4 + 12}}{6} = frac{2 pm sqrt{16}}{6} = frac{2 pm 4}{6}). So, the solutions are (x = frac{6}{6} = 1) and (x = frac{-2}{6} = -frac{1}{3}). Therefore, (1 + 2x - 3x^2 = 0) when (x = 1) or (x = -frac{1}{3}). I need to exclude these values from the domain of the original expression.Going back to the simplified expression, after canceling the common factor, I have (frac{x^2}{x}), which simplifies to (x), provided that (x neq 0) (since the denominator after canceling is (x)), and also (x neq 1) and (x neq -frac{1}{3}) as previously found.So, the inequality simplifies to (x ge -1), but I have to consider the restrictions on (x). Let me write down the simplified inequality:[x ge -1]However, I need to remember the points where the original expression is undefined: (x = -frac{1}{3}), (x = 0), and (x = 1). These points must be excluded from the solution set.So, the solution set is all real numbers (x) such that (x ge -1), excluding (x = -frac{1}{3}), (x = 0), and (x = 1). To express this in interval notation, I'll break it down into intervals separated by the excluded points:1. From (-1) to (-frac{1}{3}): This interval includes (-1) but excludes (-frac{1}{3}), so it's ([-1, -frac{1}{3})).2. From (-frac{1}{3}) to (0): This interval excludes both endpoints, so it's ((- frac{1}{3}, 0)).3. From (0) to (1): This interval excludes both endpoints, so it's ((0, 1)).4. From (1) to infinity: This interval includes all numbers greater than (1), so it's ((1, infty)).Putting it all together, the solution in interval notation is:[[-1, -frac{1}{3}) cup (-frac{1}{3}, 0) cup (0, 1) cup (1, infty)]I should double-check to make sure I haven't missed anything. Let me verify each interval:- For (x) between (-1) and (-frac{1}{3}), say (x = -frac{1}{2}), plugging into the original expression: numerator is ((-0.5)^2 + 2(-0.5)^3 - 3(-0.5)^4 = 0.25 - 0.25 - 0.375 = -0.375). Denominator is (-0.5 + 2(-0.5)^2 - 3(-0.5)^3 = -0.5 + 0.5 + 0.375 = 0.375). So, the expression is (-0.375 / 0.375 = -1), which satisfies the inequality (ge -1).- For (x) between (-frac{1}{3}) and (0), say (x = -frac{1}{4}): numerator is ((0.0625) + 2(-0.015625) - 3(0.00390625) = 0.0625 - 0.03125 - 0.01171875 = 0.01953125). Denominator is (-0.25 + 2(0.0625) - 3(-0.015625) = -0.25 + 0.125 + 0.046875 = -0.078125). So, the expression is approximately (-0.25), which is greater than (-1).- For (x) between (0) and (1), say (x = frac{1}{2}): numerator is ((0.25) + 2(0.125) - 3(0.0625) = 0.25 + 0.25 - 0.1875 = 0.3125). Denominator is (0.5 + 2(0.25) - 3(0.125) = 0.5 + 0.5 - 0.375 = 0.625). So, the expression is (0.3125 / 0.625 = 0.5), which is greater than (-1).- For (x) greater than (1), say (x = 2): numerator is (4 + 16 - 48 = -28). Denominator is (2 + 8 - 24 = -14). So, the expression is (-28 / -14 = 2), which is greater than (-1).All these test points satisfy the inequality, so my solution seems correct. I also need to check the endpoints:- At (x = -1): numerator is (1 + (-2) - 3 = -4). Denominator is (-1 + 2 - (-3) = -1 + 2 + 3 = 4). So, the expression is (-4 / 4 = -1), which satisfies the inequality (ge -1).- At (x = -frac{1}{3}), (x = 0), and (x = 1), the expression is undefined, so these points are excluded.Therefore, my final answer in interval notation is correct.

answer：Okay, so I have this problem where I need to find the sum of two polynomials, p(x) and q(x). Let me try to understand the problem step by step.First, it says that q(x) is a quadratic polynomial, which means it's a second-degree polynomial, and p(x) is a linear polynomial, so it's a first-degree polynomial. The function given is the ratio of p(x) over q(x), which is p(x)/q(x). The graph of this function has a vertical asymptote at x=1. I remember that vertical asymptotes occur where the denominator is zero but the numerator isn't zero at that point. So, this means that q(1) must be zero, but p(1) is not zero. Wait, but the problem also says p(1)=1, so p(1) is indeed not zero. That makes sense.Next, there's a hole at x=0. I recall that holes in a rational function occur when both the numerator and the denominator have a common factor, meaning that both p(x) and q(x) must have a factor of x. So, this tells me that x is a factor of both p(x) and q(x).So, putting this together, since p(x) is linear and has a factor of x, p(x) must be of the form p(x) = ax, where a is some constant. Similarly, since q(x) is quadratic and has factors of x and (x - 1) because of the vertical asymptote at x=1, q(x) must be of the form q(x) = bx(x - 1), where b is another constant.Now, the problem gives me two specific values: p(1) = 1 and q(3) = 3. Let me use these to find the constants a and b.Starting with p(1) = 1:p(1) = a*1 = a = 1So, a = 1. Therefore, p(x) = x.Next, using q(3) = 3:q(3) = b*3*(3 - 1) = b*3*2 = 6bAnd we know this equals 3, so:6b = 3Dividing both sides by 6:b = 3/6 = 1/2So, b = 1/2. Therefore, q(x) = (1/2)x(x - 1). Let me expand that to make it clearer:q(x) = (1/2)x^2 - (1/2)xNow, the problem asks for p(x) + q(x). Let's compute that:p(x) + q(x) = x + (1/2)x^2 - (1/2)xCombine like terms:The x terms: x - (1/2)x = (1 - 1/2)x = (1/2)xSo, p(x) + q(x) = (1/2)x^2 + (1/2)xLet me just double-check my work to make sure I didn't make any mistakes. 1. Identified that p(x) is linear with a factor of x, so p(x) = ax.2. Identified that q(x) is quadratic with factors of x and (x - 1), so q(x) = bx(x - 1).3. Used p(1) = 1 to find a = 1.4. Used q(3) = 3 to find b = 1/2.5. Expanded q(x) and added it to p(x), combining like terms.Everything seems to check out. I don't see any errors in my calculations or reasoning.

answer：First, identify all possible point values for each dart: 0, 2, 3, and 6 points.Next, calculate all possible totals by summing the point values of two darts:- 0 + 0 = 0- 0 + 2 = 2- 0 + 3 = 3- 0 + 6 = 6- 2 + 0 = 2- 2 + 2 = 4- 2 + 3 = 5- 2 + 6 = 8- 3 + 0 = 3- 3 + 2 = 5- 3 + 3 = 6- 3 + 6 = 9- 6 + 0 = 6- 6 + 2 = 8- 6 + 3 = 9- 6 + 6 = 12List the unique totals: 0, 2, 3, 4, 5, 6, 8, 9, 12.Count the number of unique totals: 9.

answer：Okay, so I have this problem here about an arithmetic sequence. Let me try to understand it step by step. The problem says:"Suppose that ({a_{n}}_{n geq 1}) is an increasing arithmetic sequence of integers such that (a_{a_{20}}=17) (where the subscript is (a_{20})). Determine the value of (a_{2017})."Alright, so first, let me recall what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference, usually denoted by (d). Since it's an increasing sequence, the common difference (d) must be positive.The general form of an arithmetic sequence is:[ a_n = a_1 + (n - 1)d ]where (a_1) is the first term, (d) is the common difference, and (n) is the term number.Now, the problem mentions that (a_{a_{20}} = 17). Hmm, that notation is a bit tricky. Let me parse it. The subscript of (a) is (a_{20}), which means we first find the 20th term of the sequence, and then take that term as the subscript to get another term in the sequence, which equals 17.So, let's break it down:1. First, find (a_{20}). Using the general formula:[ a_{20} = a_1 + (20 - 1)d = a_1 + 19d ]2. Then, (a_{a_{20}}) means we take the term at position (a_{20}), which is:[ a_{a_{20}} = a_1 + (a_{20} - 1)d ]But we know this equals 17, so:[ a_1 + (a_{20} - 1)d = 17 ]Substituting (a_{20}) from step 1 into this equation:[ a_1 + (a_1 + 19d - 1)d = 17 ]Let me write that out:[ a_1 + (a_1 + 19d - 1)d = 17 ]Now, let's simplify this equation. First, distribute the (d):[ a_1 + a_1 d + 19d^2 - d = 17 ]Combine like terms:- The terms with (a_1): (a_1 + a_1 d = a_1(1 + d))- The terms with (d^2): (19d^2)- The constant term with (d): (-d)So, putting it all together:[ a_1(1 + d) + 19d^2 - d = 17 ]Let me write this as:[ a_1(1 + d) + 19d^2 - d = 17 ]Now, I need to solve for (a_1) and (d). Since both (a_1) and (d) are integers, and the sequence is increasing, (d) must be a positive integer.Let me rearrange the equation to solve for (a_1):[ a_1(1 + d) = 17 - 19d^2 + d ]So,[ a_1 = frac{17 - 19d^2 + d}{1 + d} ]Since (a_1) must be an integer, the numerator (17 - 19d^2 + d) must be divisible by (1 + d). Let me denote (1 + d = k), so (d = k - 1). Substituting back into the numerator:[ 17 - 19(k - 1)^2 + (k - 1) ]Let me expand this:First, expand ((k - 1)^2):[ (k - 1)^2 = k^2 - 2k + 1 ]So,[ 17 - 19(k^2 - 2k + 1) + (k - 1) ][ = 17 - 19k^2 + 38k - 19 + k - 1 ][ = (17 - 19 - 1) + (-19k^2) + (38k + k) ][ = (-3) - 19k^2 + 39k ]So, the numerator becomes:[ -19k^2 + 39k - 3 ]And since (k = 1 + d), the denominator is (k). Therefore, we have:[ a_1 = frac{-19k^2 + 39k - 3}{k} ]Simplify this:[ a_1 = -19k + 39 - frac{3}{k} ]Since (a_1) must be an integer, (frac{3}{k}) must also be an integer. Therefore, (k) must be a divisor of 3.The divisors of 3 are 1, 3, -1, -3. However, since (d) is positive, (k = d + 1) must be greater than 1. So, possible values for (k) are 1 and 3.But (k = 1) would imply (d = 0), which contradicts the fact that the sequence is increasing (since (d > 0)). Therefore, the only possible value is (k = 3), which gives (d = 2).Now, substituting (k = 3) back into the expression for (a_1):[ a_1 = -19(3) + 39 - frac{3}{3} ][ = -57 + 39 - 1 ][ = -19 ]So, the first term (a_1) is -19, and the common difference (d) is 2.Now, we need to find (a_{2017}). Using the general formula:[ a_{2017} = a_1 + (2017 - 1)d ][ = -19 + 2016 times 2 ][ = -19 + 4032 ][ = 4013 ]So, the value of (a_{2017}) is 4013.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, confirming (a_{20}):[ a_{20} = a_1 + 19d = -19 + 19 times 2 = -19 + 38 = 19 ]Then, (a_{a_{20}} = a_{19}):[ a_{19} = a_1 + 18d = -19 + 18 times 2 = -19 + 36 = 17 ]Yes, that checks out. So, (a_{20} = 19) and (a_{19} = 17), which satisfies the condition (a_{a_{20}} = 17).Therefore, the calculations for (a_{2017}) seem correct.**Final Answer**The value of (a_{2017}) is boxed{4013}.

answer：First, I need to determine the time Deepak and his wife spend walking before they meet. They start walking in opposite directions and meet after 33 minutes. To work with the speeds given in kilometers per hour, I'll convert 33 minutes to hours.Next, I'll calculate the distance each person covers during this time. Deepak walks at 20 km/hr, and his wife walks at 13 km/hr. By multiplying their respective speeds by the time in hours, I can find out how far each has walked.Finally, since they are walking in opposite directions, the total distance they cover together when they meet is equal to the circumference of the jogging track. I'll add the distances each has walked to find the total circumference.